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# Body-Centered Cubic Unit Cell

The body-centered cubic unit cell is a cube (all sides of the same length and all face perpendicular to each other) with an atom at each corner of the unit cell and an atom in the center of the unit cell.

The unit cell completely describes the structure of the solid, which can be regarded as an almost endless repetition of the unit cell.

The volume of the unit cell is readily calculated from its shape and dimensions. This calculation is particularly easy for a unit cell that is cubic. In the case of the body-centered cubic unit cell, the atoms lying along the main diagonal of the cube are in contact with each other. Thus the diagonal of the unit cell has a length of 4 r, where r is the radius of an atom.

Atoms, of course, do not have well-defined bounds, and the radius of an atom is somewhat ambiguous. In the context of crystal structures, the diameter (2 r) of an atom can be defined as the center-to-center distance between two atoms packed as tightly together as possible. This provides an effective radius for the atom and is sometime called the atomic radius.

A more challenging task is to determine the number of atoms that lie in the unit cell. As described above, an atom is centered on each corner and in the middle of the unit cell. The atom at the center of the unit cell lies completely within the unit cell. The atoms located on the corners, however, exist partially inside the unit cell and partially outside the unit cell. In determining the number of atoms inside the unit cell, one must count only that portion of an atom that actually lies within the unit cell.

The density of a solid is the mass of all the atoms in the unit cell divided by the volume of the unit cell.

## Properties of Sodium

The virtual reality image below illustrates the body-centered cubic unit cell, which is the unit cell that describes the structure of sodium metal. The positions of the individual sodium nuclei are shown by small dots. The sodium atoms or sections of sodium atoms are shown by the spheres or sphere sections. Consult the Description of Controls or simply experiment with the features of the display.

The atomic mass of sodium is 22.9898 and the density of metallic sodium is 0.971 g/cm3.

Use the body-centered cubic unit cell to answer the following questions.

1. How many sodium atoms are contained in the unit cell?
2. What is the volume of the unit cell?
3. What is the length of each side of the unit cell?
4. What is the atomic radius of a sodium atom?
5. What is the volume of a sodium atom (based upon the atomic radius)?
6. What fraction of the volume of the unit cell is "occupied" by sodium atoms? (This fraction is the packing efficiency.)
 Body-Centered CubicUnit Cell Display Animation

This virtual reality display requires Java3D. If the display is not visible, consult the Java3D FAQ. Dragging an object with the left mouse button rotates the object. Dragging with the center mouse buttons expands the display, and dragging with the right mouse button moves the display.

Unit Cells:     Simple Cubic     Body-Centered Cubic     Face-Centered Cubic     Hexagonal Closest-Packed